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EE6-5 Solving Equations with Balances
Pages 77–78
STANDARDS
6.EE.B.5, 6.EE.B.6
Vocabulary
balance
equation
expression
sides (of an equation)
variable
Goals
Students will use pictures to model and solve equations.
PRIOR KNOWLEDGE REQUIRED
Is familiar with balances
Can solve a simple equation to find an unknown value
Can substitute numbers for unknowns in an expression
Can check whether a number solves an equation
MATERIALS
paper bags
counters
identical objects for demonstrations (see below)
a pan balance
connecting cubes
a paper bag for each pair of students
a ruler, masking tape, or string
NOTE: You will need several identical objects for demonstrations throughout
this lesson. The objects you use should be significantly heavier than a
paper bag, so that the presence of a paper bag on one of the pans of the
balance does not skew the pans. Apples are used in the lesson plan below
(to match the pictures in the AP Book), but other objects, such as small fruit
of equal size, metal spoons, golf balls, tennis balls, or cereal bars will work
well. If a pan balance is not available, refer to a concrete model, such as
a seesaw, to explain how a pan balance works, and use pictures or other
concrete models during the lesson.
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Review pan balances. Show students a pan balance. Place the same
number of identical (or nearly identical) apples on both pans, and show
that the pans balance. Remind students that when the pans, or scales, are
balanced, it means there is the same number of apples on each pan.
Removing the same number of apples from each pan keeps them
balanced. Place some apples in a paper bag and place it on one pan, then
add some apples beside the bag. Place the same total number of apples
on the other pan. ASK: Are the pans balanced? (yes) What does this mean?
(the same number of apples is on each pan) Take one apple off each pan.
ASK: Are the pans still balanced? Repeat with two apples. Continue removing
the same number of apples from each pan until one pan has only the bag
with apples on it. ASK: Are the pans balanced? Can you tell how many
apples are in the bag? Show students the contents of the bag to check their
answer. Repeat the exercise with a different number of apples in the bag.
Expressions and Equations 6-5
D-13
Solving addition equations given by a balance model. Divide a desk
in half (you can use a ruler, string, or masking tape) and explain that the
parts on either side of the dividing line will be the pans. Ask students to
imagine that the pans are balanced. As you did above, place a paper bag
with apples in it along with some other apples on one side of the line, and
place the same number of apples (altogether) on the other side of the line.
Ask students how many apples need to be removed from both sides of the
balance to find out how many apples are in the bag. Students can signal
their answer. Remove the apples, then ask students to tell how many apples
are in the bag. Show the contents of the bag to check the answer. Repeat
with a different number of apples.
ACTIVITY 1
Students can work in pairs to create models of addition equations and
solve them. Each pair will need connecting cubes and a paper bag.
They can use a ruler, masking tape, string, or the line along which
desks meet as the dividing line for their model. Partner 1 places some
cubes in a paper bag and some more cubes beside it on one side of
the line. Partner 1 then places an equal (total) number of cubes on the
other side of the line. Partner 2 has to determine how many cubes are
in the bag. Partners switch roles and repeat.
The “pans” balance each
other. The numbers on
both sides are equal.
Create more such models and have students write the equation for each
one. After you have done a few models that follow this pattern, start placing
the bag on different sides of the line, so that students have to write the
expressions with unknown numbers on different sides of the equation.
Solving addition equations using the balance model. Return to the model
that corresponds to the equation x + 2 = 7. ASK: What do you need to do
to find out how many apples are in the bag? (remove two apples from each
side) Invite a volunteer to remove the apples, then have students write both
the old equation and the new one (x = 5), one below the other. Repeat with
a few different examples.
ASK: What mathematical operation describes taking the apples away?
(subtraction) Write the subtraction for each of the equations above
vertically. Example:
D-14
Teacher’s Guide for AP Book 6.1
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x+2=7
Writing equations from a balance model. Remind students that we can
use variables to represent numbers we do not know. Place a paper bag
with 5 apples and 2 more apples on one side of the line, and place 7 apples
on the other side of the line. Ask students to write an expression for the
number of apples on the side with the paper bag. Explain that an equation
is like a pair of balanced pans (or scales), and the equal sign shows that
the number of apples on each pan is the same. Remind students that the
parts of the equation on either side of the equal sign are called the sides of
the equation. Each pan of the balance becomes a side in the equation and
the “balance” on the desk becomes x + 2 = 7.
x+2 = 7
−2 −2
ASK: How many apples are left on the right side of the equation? (5)
What letter did we use to represent the number of apples in the bag?
(x) Remind students that we write this as “x = 5.” Repeat with the other
equations used earlier.
Solving addition equations without using the balance model. Present a
few equations without a corresponding model. Have students signal how
many apples need to be subtracted from both sides of the equation, and
then write the vertical subtraction for both sides.
Exercises
a) x + 5 = 9
b) n + 17 = 23
c) 14 + n = 17
d) p + 15 = 21
Students who have trouble deciding how many apples to subtract without
drawing a model can complete the following problems.
Exercises: Write the missing number.
a) x + 15
b) x + 55
c) x + 91
- 15 x x x Bonus: x + 38
38
Finally, give students a few equations and have them work through the
whole process of subtracting the same number from both sides to find
the unknown number. Exercises:
a) x + 5 = 14
b) x + 9 = 21
c) 2 + x = 35
d) x + 28 = 54
Sample solution:
a) x + 5 = 14
−5
−5
x = 9
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Bonus: The scales in the margin are balanced. Each bag has the same
number of apples in it. How many apples are in the bag? Hint: You can
cross out whole bags too!
(MP.3)
Solving multiplication equations given by a model. Divide a desk into
two parts and place 3 bags (with 4 cubes in each) on one side of the line
and 12 separate cubes on the other side. Tell students that the “pans” are
balanced. What does this say about the number of cubes on both pans?
(they are equal) How many cubes are on the pan without the bags? (12)
How many cubes are in the bags in total? (12) How many cubes are in each
bag? (4) How do you know? (divide 12 into 3 equal groups, 12 ÷ 3 = 4)
Invite a volunteer to group the 12 cubes into 3 equal groups to check the
answer. Show students the contents of the bags to confirm the answer.
Repeat the exercise with 4 bags and 20 cubes, 5 bags and 10 cubes,
2 bags and 6 cubes. Students can signal the number of cubes in one
bag each time.
Expressions and Equations 6-5
D-15
Writing equations from models. Remind students that the pans of the
balance become the sides of an equation, and that the equal sign in the
equation shows that the pans are balanced. If students have, say, 3 bags
with the same number of cubes in each, they write the total number of
cubes in the bags as 3 × b. Present a few equations in the form of a model,
and have students write the corresponding equations using the letter b for
the unknown number.
Exercises
a)
b)
Now have students use the models to solve the equations. Show them how
to write the solution below the equation and have students record their
solutions. Example:
2 × b = 12
b=6
ACTIVITY 2
(MP.2)
Drawing models to solve equations. Tell students that the next task will
be the opposite of what they have been doing: now they will start with an
equation and draw a model for it. Remind students that when we draw
pictures in math class it is most important to draw the correct numbers
of objects. Shading, color, and other artistic features or details are not
important. Our drawings in math should be simple and we shouldn’t spend
too much time on them. Demonstrate making a simple drawing of a pan
balance, and remind students that they can use circles, squares, or big
dots for cubes and boxes for paper bags. Exercises: Draw models and
use them to solve the equations.
a)3 × b = 15
b) 4 × b = 8 c) 9 × b = 18
d) 8 × b = 24
Using division to find the missing factor. ASK: Which mathematical
operation did you use to write an equation for each balance? (multiplication)
Which mathematical operation did you use to find the number of apples in
each bag? (division) Have students show the division in the models they
have drawn by circling equal groups of dots. For example, in Exercise a)
above they should circle three equal groups of dots. ASK: What number
do you divide by? (the number of bags)
D-16
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Students can work in pairs to create models of multiplication equations
and solve them. Each pair will need connecting cubes and several
paper bags. They can use a ruler, masking tape, string, or the line
along which desks meet as the dividing line for their model. Partner 1
places the same number of cubes in several paper bags on one side
of the line, and places the number of cubes equal to the total in the
bags on the other side. Partner 2 has to write the equation, tell how
many cubes are in each bag, and record the solution. Partners switch
roles and repeat.
Rewriting equations so that the missing number is the first factor.
Remind students that order does not matter in multiplication. Have students
give you a few examples and write them on the board (Example: 3 × 4 = 12
and 4 × 3 = 12, so 3 × 4 = 4 × 3). Remind students that letters represent
numbers that you do not know, so anything that works with numbers will
work with letters too. Then write:
b×3=
Ask students what this expression will be equal to. (3 × b) Repeat with a
few other products where the unknown comes first.
Write the following equations on the board:
a)3 × b = 18
b) 4 × b = 16
c) 9 × b = 18
d) 8 × b = 21
Have students rewrite them so that the variable is the first factor. Ask
students how they can solve these equations. Have students make a
model to find the answers.
Multiplying and dividing by the same number does not change the
starting number. Have students solve the following questions:
a)(5 × 2) ÷ 2
b) (3 × 2) ÷ 2
c) (8 × 2) ÷ 2
d)(5 × 4) ÷ 4
e) (9 × 3) ÷ 3
f) (10 × 6) ÷ 6
SAY: Look at the questions you solved. How are they all the same? (you
start with a number, then multiply and divide by the same number) ASK:
Did you get back to the same number you started with? (yes) Does it matter
what number you started with? Does it matter what number you multiplied
and divided by as long as it was the same number? (no) Have students
write their own question of the same type, swap questions with a partner,
and check that the answer is the number they started with.
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Illustrate the same principle using concrete materials: Show students a train
of 4 connecting cubes. Tell students that you want to multiply that train by
a number, say 3. What will the answer look like? (3 trains of 4 cubes) Make
two more trains of 4 cubes. Then say that you want to divide the result by 3.
What will the result of the division look like? (1 train) Explain that you can do
the same with unknown numbers: Show one paper bag with some cubes
and say that you want to multiply it by 3. What will the answer look like?
(3 such bags) Now you want to divide the result by 3. What will you get?
(1 bag again)
Finally, write on the board:
(
× 3) ÷ 3 =
ASK: What will we get when we perform the multiplication and the division?
(the box) Repeat with equations that include letters as variables:
a)(b × 3) ÷ 3
Expressions and Equations 6-5
b) (b × 5) ÷ 5
c) (b × 6) ÷ 6
d) (b × 10) ÷ 10
D-17
Solving equations by dividing both sides by the same number. Write the
questions below and have students signal the number they would divide
the product by to get back to b.
(b × 7) ÷
= b(b × 2) ÷
= b(b × 4) ÷
=b
(b × 8) ÷
= b(b × 12) ÷
= b(b × 9) ÷
=b
If available, show students a pan balance with 3 bags of 5 apples on one
pan and 15 apples on the other pan. Invite a volunteer to write the equation
for the balance: 3 × 5 = 15. ASK: How many apples are in one bag? (5)
Have a volunteer make three groups of 5 apples on the side without the
bags. Point out that there are three equal groups of apples on both sides of
the balance. Remove two of the bags from one side, and two of the groups
from the other side. SAY: I have replaced three equal groups on each side
with only one of these groups. What operation have I performed? (division by
3) Are the scales still balanced? (yes) Point out that when you perform the
same operation on both sides of the balance, the scales remain balanced.
What does that mean in terms of the equation? Write the equation that
shows the division below the original equation:
b × 3 ÷ 3 = 15 ÷ 3
Have students calculate the result on both sides (b on the left side, 5 on
the right side). Write on the board b = 5 (align the equal signs vertically).
Demonstrate that the bags indeed contain 5 apples. Repeat with a few
more examples.
Finally, have students solve equations by dividing both sides of the
equation by the same number.
a) b × 7 = 21
b) b × 2 = 12
c) b × 4 = 20
d) b × 6 = 42
e) b × 3 = 27
f) b × 9 = 72
Bonus
g)3 × b = 270
h) 8 × b = 4,000
i)7 × b = 42,000
j) 6 × b = 720,000
Sample solution:
a) b × 7 = 21
b × 7 ÷ 7 = 21 ÷ 7
b=3
Answers: b) 6, c) 5, d) 7, e) 9, f) 8, Bonus: g) 90, h) 500,
i) 6,000, j) 120,000
D-18
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Exercises
EE6-6 Solving Equations—Guess and Check
Page 79
STANDARDS
6.EE.B.5
Goals
Students will solve equations of the form ax + b = c by guessing small
values for x, checking by substitution, and then revising their answer.
Vocabulary
equation
expression
solving for a variable
PRIOR KNOWLEDGE REQUIRED
Can read tables
Can substitute numbers for variables in equations
Can check whether a number solves an equation
MATERIALS
paper bags
counters
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x 4x + 3
1
7
2
11
3
15
4
19
5
23
6
27
7
31
Is equation
true?
Introduce using a table to solve equations. Draw a model on the board
and build the corresponding equation:
There are x counters in each bag.
There are 4x counters in all the bags because there are 4 bags.
There are 4x + 3 counters altogether.
Make the same model using 4 bags and 31 counters. Put 3 outside the
bags and put 1 counter at a time in each bag until all 31 are used. ASK:
What equation can we write for this model? (4x + 3 = 31) Show students
how to solve 4x + 3 = 31 by using a table and substituting different values
of x in sequence (x = 1, x = 2, x = 3, and so on) into the expression on the
left side of the equation (see margin).
Point out the connection between the table and the method using counters:
In the table, each time we increase the value of x by 1, it is as though
we are adding a counter to each bag (4 counters for 4 bags) and checking
how many counters are used in total. We stop when we see that all
31 counters are used.
h 7h + 2
5
37
6
44
Is equation
true?
Introduce the guess and check method to solve equations. Show the
equation 7h + 2 = 44. Tell students that you are going to solve this equation
by guessing and checking. Start by guessing h = 5. ASK: If h = 5, what is
7h + 2? (37) Should h be higher or lower to make 7h + 2 = 44? (higher)
What would your next guess be? (6) If h = 6, what is 7h + 2? (44) Is the
equation true? (yes) Draw the table at left on the board and SAY: h = 6
makes the equation 7h + 2 = 44 true, so h = 6 is the answer.
Compare the two methods of solving equations. ASK: Which method
requires less work? Which method is quicker? (the guess and check method
is quicker) Which method is more like looking up a word in the dictionary
using alphabetical order? (guess and check) Which method is more like
Expressions and Equations 6-6
D-19
looking up a word in the dictionary without knowing or using alphabetical
order? (using the table) Have students explain the connection. (In a dictionary,
the words at the top of each page you turn to tell you whether to look to the
right or to the left; they tell you if you have gone too far or not far enough.)
Exercises
a)Replace x with 5 and say whether 5 is too high or too low.
i)4x + 1 = 25
ii) 5x + 3 = 23
iii) 2x + 4 = 16
x 4x + 1 Answer x 5x + 3 Answer x 2x + 4 Answer
5
5
5
b)Use the answers in part a) to try a higher or lower number and solve
each equation.
Answers: a) i) 21, too low, ii) 28, too high, iii) 14, too low, b) x = 6 works,
ii) x = 4 works, iii) x = 6 works
Extensions
(MP.1, MP.7)
1.
How many digits does the solution to 3x + 5 = 8,000 have? Explain.
Hint: 1-digit numbers are between 1 and 9, 2-digit numbers are between
10 and 99, and so on.
Solution: To determine the number of digits in the solution, we need to
determine the first power of 10 (10, 100, 1,000, etc.) that is greater than
the solution. We can substitute increasing powers of 10 for the variable
until the answer is larger than 8,000:
3(10) + 5 = 35
3(100) + 5 = 305
3(1,000) + 5 = 3,005
3(10,000) + 5 = 30,005
(MP.7)
2.
How many solutions can you find to 2x + 1 = 4y − 1 if x and y
are whole numbers?
Solution: Find 2x + 1 for various values of x:
x
1
2
3
4
5
2x + 1 3
5
7
9 11
Now find 4y - 1 for various values of y:
y
1
4y - 1 3
D-20
2
3
4
5
7 11 15 19
Teacher’s Guide for AP Book 6.1
COPYRIGHT © 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION
So x is between 1,000 and 10,000, which means that it has 4 digits.
(Indeed, x = 2,665.)
Look for numbers that are the same in the second rows:
2x + 1 = 3 = 4y − 1 when x = 1 and y = 1
2x + 1 = 7 = 4y − 1 when x = 3 and y = 2
2x + 1 = 11 = 4y − 1 when x = 5 and y = 3
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Students might continue the pattern to find more solutions
(x = 7 and y = 4 is the next one).
Expressions and Equations 6-6
D-21